Note that, as opposed to the “classical" majorization concepts of Section REF , a definition of \((a,b)\) -majorization cannot be given in terms of partial sums of non-increasing, non-decreasing, or absolute orders. Similar generalizations of majorization such as \(d\) -majorization [1]} are introduced via an alternative definition of majorization involving doubly stochastic matrices (see also Chapter 2 and Section 14.B of [2]}) or in terms of optimality for particular classes of separable convex objective functions (see, e.g., [3]}). Since we focus in this article on optimization problems, we follow the latter approach to define and interpret \((a,b)\) -majorization.
